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Bifurcations of traveling wave solutions for a two-component Fornberg-Whitham equation. (English) Zbl 1232.35125

The authors study a two-component Fornberg-Whitham equation given by \(u_t=u_{xxt}-u_x-uu_x+3u_xu_{xx}+uu_{xxx}+\rho_x\), \(\rho_t=-(\rho u)_x\), where \(u=u(x,t)\) is the height of the water surface above a horizontal bottom, and \(\rho=\rho(x,t)\) is related to the horizontal velocity field. Under additional conditions it is shown that there are smooth solutions, non smooth solutions and periodic wave solutions. The proofs are based on transforming the Fornberg-Whitham system into a planar dynamical system and on a discussion of phase portraits. Moreover, the authors present some explicit solutions.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
35C07 Traveling wave solutions
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B65 Smoothness and regularity of solutions to PDEs
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[1] Lenells, J., Traveling wave solutions of the Camassa-Holm equation and Korteweg-de Vries equations, J Nonlinear Math Phys, 11, 508-520 (2004) · Zbl 1069.35072
[2] Camsssa, R.; Holm, D. D.; Hyman, J. M., An integrable shallow water equation with peaked solitons, Phys Rev Lett, 71, 1661-1664 (1993) · Zbl 0972.35521
[3] Fornberg, B.; Whitham, G., A numerical and theoretical study of certain nonlinear wave phenomena, Phil Trans R Soc Lond A, 289, 373-404 (1978) · Zbl 0384.65049
[4] Zhou, J.; Tian, L., Solitons, peakons and periodic cusp wave solutions for the Fornberg-Whitham equation, Nonlinear Anal Real World Appl, 11, 1, 356-363 (2010) · Zbl 1181.35222
[5] Zhou, J.; Tian, L., A type of bounded traveling wave solutions for the Fornberg-Whitham equation, J Math Anal Appl, 346, 1, 255-261 (2008) · Zbl 1146.35025
[6] Wazwaz, A.-M., Peakons, kinks, compactons and solitary patterns solutions for a family of Camassa-Holm equations by using new hyperbolic schemes, Appl Math Comput, 182, 1, 412-424 (2006) · Zbl 1106.65109
[7] Zhou, J.; Tian, L., Periodic and solitary wave solutions to the Fornberg-Whitham equation, Math Prob Eng (2009) · Zbl 1180.35450
[8] Chen, M.; Liu, S.-Q.; Zhang, Y., A two-component generalization of the Camassa-Holm equation and its solutions, Lett Math Phys, 75, 1, 1-15 (2006) · Zbl 1105.35102
[9] Ivanov, R., Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46, 6, 389-396 (2009) · Zbl 1231.76040
[10] Constantin, A.; Ivanov, R. I., On an integrable two-component Camassa-Holm shallow water system, Phys Lett A, 372, 48, 7129-7132 (2008) · Zbl 1227.76016
[11] Chen, A.; Li, J.; Deng, X.; Huang, W., Travelling wave solutions of the Fornberg-Whitham equation, Appl Math Comput, 215, 8, 3068-3075 (2009) · Zbl 1195.35089
[12] Yin, J.; Tian, L.; Fan, X., Classification of travelling waves in the Fornberg-Whitham equation, J Math Anal Appl, 368, 1, 133-143 (2010) · Zbl 1189.35042
[13] Jiang, B.; Bi, Q., Smooth and non-smooth traveling wave solutions of the Fornberg-Whitham equation with linear dispersion term, Appl Math Comput, 216, 7, 2155-2162 (2010) · Zbl 1192.65135
[14] Tian, L.; Gao, Y., The global attractor of the viscous Fornberg-Whitham equation, Nonlinear Anal Theory Methods Appl, 71, 11, 5176-5186 (2009) · Zbl 1181.35027
[15] Meng, Y.; Tian, L., Boundary control on the viscous Fornberg-Whitham equation, Nonlinear Anal Real World Appl, 11, 2, 827-837 (2010) · Zbl 1180.35357
[16] Zhou, J.; Tian, L.; Fan, X., Soliton, kink and antikink solutions of a 2-component of the Degasperis-Procesi equation, Nonlinear Anal Real World Appl, 11, 4, 2529-2536 (2010) · Zbl 1196.35196
[17] Perko, L., Differential equations and dynamical systems (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0717.34001
[18] Li, J. B.; Dai, H. H., On the study of singular nonlinear travelling wave equations: dynamical approach (2007), Science Press: Science Press Beijing
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